Around Efimov’s differential test for homeomorphism

Abstract

In 1968, N.\,V.~Efimov proved the following remarkable theorem: \textit{Let $f:\mathbb{R}^2\to\mathbb{R}^2\in C^1$ be such that $\det f'(x)<0$ for all $x\in\mathbb{R}^2$ and let there exist a function $a(x)>0$ and constants $C_1\geqslant 0$, $C_2\geqslant 0$ such that the inequalities $|1/a(x)-1/a(y)|\leqslant C_1 |x-y|+C_2$ and $|\det f'(x)|\geqslant a(x)|\operatorname{curl}f(x)|+a^2(x)$ hold true for all $x, y\in\mathbb{R}^2$. Then $f(\mathbb{R}^2)$ is a convex domain and $f$ maps $\mathbb{R}^2$ onto $f(\mathbb{R}^2)$ homeomorphically.} Here $\operatorname{curl}f(x)$ stands for the curl of $f$ at $x\in\mathbb{R}^2$. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.